Clinical statistics for non-statisticians: Day one
Start with a bad joke
Two statistics are sitting in a bar. One turns to the other and asks, “So, how do you like married life?”
The other statistic responds …
Put your reaction (“Ha ha”, “Groan”, etc.) in the chat box.
Before I begin anything important, I like to start with a silly joke. Now on Zoom, I often miss student reactions. So when I say something funny, I want you to type “Ha ha” or “Smile” or “LMFAO”. The acronym LMFAO means laughing my something … I forget how the rest of it goes.
Now if the joke is corny, like a really really bad pun, it’s okay to put “Groan”. The only thing bad is if I tell a joke and get no reaction at all.
I’ll be sneaking in some jokes throughout the talk and I really want a reaction from you, good or bad. If I don’t get any reaction to a bad pun, your “pun”ishment will be more bad puns.
So here’s the joke. It has been floating around on the Internet for quite a while, and I can’t find the person who gets credit for this. But here goes.
[Read joke and finish with] “It’s okay but you lose a degree of freedom.”
Okay, I’m waiting for reactions.
Introduction
Tell us one interesting number about yourself
Examples
I have traveled to eight countries outside the United States
(Canada, Italy, China, France, Russia, England, Holland, and Iceland)
I did not learn how to drive until I was 29 years old
My highest chess rating was 1802, but I am not that good any more.
Speaker notes
I want to learn a bit about all of you, and I’m going to do this in a statistical way. Tell me three numbers about yourself. These could be something simple, like the number of children you have or something exotic like the height of the highest mountain you have climbed.
Here are three numbers about me.
A bit more about myself
PhD in Statistics in 1982 from the University of Iowa
Currently full professor
Part-time statistical consultant
Funded on 18 research grants
Over 100 peer-reviewed publications
Website with over 2,000 pages
Many invitations to talk at conferences
I have a PhD in Statistics from the University of Iowa. I have always had a strong interest in the computational side of Statistics. My dissertation was 150 pages, and 100 of those pages were computer generated graphs.
I am currently a full professor at the University of Missouri-Kansas City in the Department of Biomedical and Health Informatics. I also do statistical consulting on a part-time basis.
I have been a prolific researcher, receiving support from 18 different grants, and writing over 100 peer-reviewed publications.
I started a website in 1998, writing about data analysis, research ethics, and evidence based medicine. I wrote about two or three pages every week and my site now has over 2,000 pages. It shows the value of persistence.
I love to talk about Statistics and have given many presentations at regional, national, and international conferences. This ranges from short 15 minute talks to day long short courses.
Outline of the three day course
Day one: Numerical summaries and data visualization
Day two: Hypothesis testing and sampling
Day three: Statistical tests to compare treatment to a control and regression models
My goal: help you to become a better consumer of statistics
Day one topics
Numerical summaries
When should you present the mean versus the median
When should you present the range versus standard deviation
How should you display percentages
Why should you round liberally
Day one topics (continued)
Data visualization
How should you display continuous data
Why is the normal bell-shaped curve important
How should you display categorical data
How do you illustrate trends and patterns
What are some common mistakes in the choice of colors
Counting and proportions
Counts are the most common statistic
Counts are error prone
Counts require a solid operational definition
Speaker notes
Let’s start with the simplest statistic of all a simple count. This is probably the most common statistic produced.
But counts can be tricky. The counting process is error prone and requires a solid operational definition.
Student exercise
Count the number of occurrences of the letter “e”.
A quality control program is easiest
to implement from the top down.
Make sure that you understand the
the commitment of time and money
that is involved. Every workplace is
different, but think about allocating
10% of your time and 10% of the
time of all your employees to
quality control.
Speaker notes
Here’s an exercise I want you to do. Just count the number of occurrences of the letter “e”. Once you have your answer, type it in the chat box.
[Pause here]
The numbers are different because of two things. First, it is easy to make mistakes. Did anyone notice the repetition of the word “the” at the end of the third line and the beginning of the fourth. It would be easy to miss that and count one less “e”.
What did you do with the first e in “Every”?
Did you count the e’s in the quotes itself or also on the slide instructions and the slide header?
Counting sperm
Figure 1: Image of a haemocytometer
Speaker notes
This image is take from the WHO laboratory manual for the examination and processing of human semen, published in 2021. It shows a haemocytometer, an instrument used for counting the number of cells. To get a proper count, you need to include any cells inside the four by four grid of large squares in the middle of this micrograph. But what does “inside” mean? Should you count only those cells entirely inside the four by four grid. Or should you include cells that are partially inside the grid?
One rule is to count cells if the head of the sperm cell touches the top or right side of a square, but not if it touches the bottom or left side of the square. And don’t count a sperm cell if only the tail is inside the square.
That’s not the only way you can do this, but just make sure that whatever convention you use for deciding “inside” versus “outside” is consistent across your laboratory.
Tables of counts, using the Titanic data.
Figure 2: Counts of survival by gender
Percentages dividing by column totals
Figure 3: Column percentages
Percentages dividing by row totals
Row percentages
Percentages divided by grand total
Cell percentages
My recommendations
Treatment or exposure as rows
Outcome as columns
Usually report row percentages
Female mortality rate: 33%
Male mortality rate: 83%
But sometimes column percentages
Survivors: 68% female, 32% male
Some rationale for these choices
My way
Survived
No Yes
Sex Female 33% (154) 67% (308)
Male 83% (863) 17% (142)
Not my way
Sex
Female Male
Survived No 33% (154) 83% (863)
Yes 67% (308) 17% (142)
Speaker notes
Now, I believe it is important to think carefully about which is your rows and which is your columns. Here’s the layout that I recommend on the left and the layout that I don’t recommend on the right. The key comparison is among survival rates, 67% for females and only 17% for males. When you orient my way with the treatment/exposure (Sex) as rows and the outcome (Survived) as the columns, the numbers 67% and 17% are very close to one another. In the alternate layout the numbers you are most interested in comparing are not as close together.
Now this is not an absolute rule. Sometimes I’ll switch things up. But about 90% of the time, I find that the layout with the treatment or exposure as the rows and the outcome as the columns, the table just looks better.
On your own
Calculate row and column percentages for the following tables. Interpret your results.
Speaker notes
Now try to report both column and row percents for one of these two tables. Breakout room #1 work on the passenger class table and breakout room #2 work on the child data.
Put your percentages in a table using a word processing program or text editor so you can share your results with the group.
Be sure to interpret these numbers. Come back together again in about 10 minutes.
The mean (average)
Figure 6: Cartoon image of Professor Mean
Speaker notes
Here’s a cartoon image of Professor Mean. I know this looks like it was drawn by a professional artist, but it was actually drawn by me. Really!
Professor Mean is my alter ego on the Internet. For those who don’t get the inside joke, I point out that Professor Mean is not just your average professor.
I will use the terms mean and average interchangeably througout this talk.
Bacteria before and after A/C upgrade
Room Before After Change
121 11.8 10.1 -1.7
125 7.1 3.8 -3.3
163 8.2 7.2 -1.0
218 10.1 10.5 0.4
233 10.8 8.3 -2.5
264 14 12 -2.0
324 14.6 12.1 -2.5
325 14 13.7 -0.3
Use of the mean for ordinal data
Gould 1985
Figure 8: Gould 1985
Speaker notes
Stephen Jay Gould was a famous Evolutionary Biologist. He was a prolific writer with 20 books and 300 essays. Much of his writing was for academic researchers, but just as much was for the general public.
One of his most famous essays was “The Median Isn’t the Message”. The title is a take-off of a quote by Marshall McLuhan, “The medium is the message” which itself has an interesting history that you should investigate on your own.
The Gould essay was written in 1985 for Discover Magazine. It has been reprinted many times, and you can easily find the full text with a simple Google search.
The image shown here is taken from phoenix5.org, an informational site for patients with prostate cancer.
Bridge 2001, PMID: 11405531
Figure 9: Bridge and McKenzie 2001
Bridge 2001, PMID: 11405531 (continued)
The measurement of airway resistance by the interrupter technique (Rint) needs standardization. Should measurements be made be during the expiratory or inspiratory phase of tidal breathing? In reported studies, the measurement of Rint has been calculated as the median or mean of a small number of values, is there an important difference?
Bridge 2001, PMID: 11405531 (continued)
In the present data the mean of a set of values contributing to a measurement was not significantly different from the median. However, the use of the median has been recommended since it is less affected by possible outlying values such as might be included by fully automated equipment.
Chen 2019, PMID: 31806195
Figure 10: Chen et al 2019
Chen 2019, PMID: 31806195 (continued)
Background: The prices of newly approved cancer drugs have risen over the past decades. A key policy question is whether the clinical gains offered by these drugs in treating specific cancer indications justify the price increases.
Chen 2019, PMID: 31806195 (continued)
Results: We found that between 1995 and 2012, price increases outstripped median survival gains, a finding consistent with previous literature. Nevertheless, price per mean life-year gained increased at a considerably slower rate, suggesting that new drugs have been more effective in achieving longer-term survival. Between 2013 and 2017, price increases reflected equally large gains in median and mean survival, resulting in a flat profile for benefit-adjusted launch prices in recent years.
Percentiles
Figure 11: Illustration of the 75th percentile
Speaker notes
I want to mention percentiles briefly. A percentile is a value that splits the data so that a certain percentage is smaller and a certain percentage is larger.
The 75th percentile, for example will be above 75% of the data and below 25% of the data. This graph illustrates the 75th percentile for some arbitrary data. THe gray bars represent about 75% of the data and the white bars represent about 25% of the data.
I use a few weasel words like “roughly” and “about” because you can’t always get a perfect split. But you can usually come close.
Computing percentiles
Many formulas
Differences are not worth fighting over
My preference (pth quantile)
Sort the data
Calculate p*(n+1)
Is it a whole number?
Yes: Select that value, otherwise
No: Go halfway between
Special cases: p(n+1) < 1 or > n
Speaker notes
There are close to a dozen different ways to compute a percentile, but the differences between the values selected are small and not worth fussing about.
Here is my preference for choosing the pth quantile (remember that for quantiles, you range between 0 and 1, not between 0 and 100).
Calculate the quantity p*(n+1). If that value is a whole number, great! You just select that value. If it is a fractional value, round up and down and go halfway between.
Once in a while, you’ll get an extreme case, where p(n+1) is less than 1 or greater than n. Just use a bit of common sense.
If you have nine values and p(n+1) is 9.2, you can’t go halfway between the 9th and 10th observations. There is no 10th observation. So just choose the 9th or largest value.
Likewise if p(n+1) is 0.8, you can’t go halfway between the zeroth and first observation. There is no zeroth observation. Just choose the first or smallest value.
Some examples of percentile calculations
Example for n=39
For 5th percentile, p(n+1)=2 -> 2nd smallest value
For 4th percentile, p(n+1)=1.6 -> halfway between two smallest values
For 2nd percentile, p(n+1)=0.8 -> smallest value
Speaker notes
Suppose you have 39 observations. For the 5th percentile or the 0.05 quantile, p(n+1) equals 2. Lucky you. The second smallest observation is the 5th percentile. For the 4th percentile or the 0.04 quantile, you get p(n+1) equal to 1.6. Go halfway between 1, the smallest value, and 2, the second smallest value.
The 2nd percentile represents one of the special cases. You calculate p(n+1) and get 0.8. You can’t go halfway between 0 and 1, so just choose the smallest value.
Some terminology
Percentile: goes from 0% to 100%
Quantile: goes from 0.0 to 1.0
90th percentile = 0.9 quantile
Quartiles: 25th, 50th, and 75th percentiles
Lower quartile: 25th percentile
Upper quartile: 75th percentile
Speaker notes
A percentile always refers to a percentage. So it has to be between 0% and 100%. Sometimes, you may see references to a quantile. A quantile is a percentile, but is expressed as a proportion rather than a percent. A quantile goes from 0.0 to 1.0. The 25th percentile and the 0.25 quantile are the same thing.
You might see the term “quartiles”. These are the 25th, 50th, and 75th percentiles. These three values split the data into quarters.
If you see “lower quartile”, it means the 25th percentile. Likewise, “upper quartile” means the 75th percentile.
Let me be try to be careful about terminology here. But, sometimes I will mess up and use “percentile” when I mean “quantile”.
When you should use percentiles
Characterize variation
Exposure issues
Not enough to control median exposure level
Quantify extremes
What does “upper class” mean?
Quality control
Almost all products must meet a minimum standard
Speaker notes
There are many reasons why you might be interested in percentiles rather than the mean or median. Actually, the median is a percentile, the 50th percentile, but what I mean is percentiles other than 50%.
One important use of percentiles is looking at the middle 50% of the data. This is the data between the lower quartile (25th percentile) and the upper quartile (75th percentile). Is the middle 50% of the data bunched tightly together or spread widely apart?
Percentiles are also important in the study of exposures. If you work in an environment where the median worker has a safe level of exposure, you could easily end up with 20%, 30% or more of the workers dying from unsafe exposures. It is important to insure that not just the median, but a very high percentile like the 99th percentile of exposure levels is at a safe level.
Percentiles also help to define extreme groups. You can, for example, define the term upper class as anyone earning more than the 90th percentile of income.
Percentiles also can help with quality control. If you make a claim about a product, you want to make sure that that claim is not valid at a median level but at a much higher level. You don’t sell 500 mg bottles of liquid Tylenol is your factory is churning out a median fill level of 500 mg. Half of your customers would be cheated. Instead you insure that the 98th percentile coming out of the factory floor is at least 500 mg. You lose a bit of money because most bottles contain more than 500 mg, but the cost of an irate customer is worth more than the cost of 50 overfilled bottles.
Standard deviation
\[S = \sqrt{\frac{1}{n-1}\Sigma(X_i-\bar{X})^2}\]
At least one alternative formulas.
Speaker notes
The standard deviation is a commonly used measure of how spread out the data is. The formula is a bit messy, but if you look carefully at it, you will see that it is a measure of how far each individual value is from the overall mean.
Now, maybe you’ve seen or used a different formula. Don’t worry about it. In a short course like this, I won’t ask you to calculate anything as tedious as a standard deviation. Let the computer do all of the work.
Why is variation important
Variation = Noise
Too much noise can hide signals
Variation = Heterogeneity
Too little heterogeneity, hard to generalize
Too much heterogeneity, mixing apples and oranges
Variation = Unpredictability
Too much unpredictability, hard to prepare for the future
Variation = Risk
Too much risk can create a financial burden
Speaker notes
I want to discuss measures of variation now. Variation gets at the heart and soul of clinical statistics. A large portion of statistical analysis involves characterizing variation.
Variation can be thought of as a measure of noise. In general, but not always, noise is bad. Consider measuring a patient’s glucose level, to see if you have early evidence of diabetes. Your glucose level varies a lot during the day based on whether you skipped breakfast or decided to get a mid-afternoon Snickers bar. Your glucose level is noisy. A high level might or might not mean trouble. A low value might or might not mean you are safe. The large standard deviation of your measures of blood glucose indicates noise.
That’s why you are asked to take an overnight fast before testing your blood glucose level. Controlling your diet by not eating anything after midnight provides a more consistent measure of blood glucose. It has a smaller standard deviation and a high or low value is more helpful in diagnosis.
Variation can also be thought of as a measure of heterogeneity. Heterogeneity is also bad sometimes, but there are times when you want a fair amount of heterogeneity. A research study that has a lot of variation is better at providing a complete picture of what a typical patient is. Outcomes that are consistent in the presence of demographic heterogeneity give you more confidence in generalizing the results of a research study. You have some assurance that the therapy is not restricted to helping a small segment of patients.
Too much heterogeneity, though, can mean that any summary measure is a mixture of apples and oranges. You have to find the right balance.
Variation can be equated to unpredictability. The number of beds needed in a hospital does vary, and this makes it difficult to staff properly. The more variation in beds needed, the more headaches you have.
Variation can also be equated to risk. If you invest in a new drug, paying millions or even billions of dollars in testing, you are doing so with the hope that your investment will pay off. Unfortunately, the market for your drug is uncertain, and you might end up with no market at all if your clinical trials fail to convince FDA. There is variation in the return on your investment, and the more variation there is, the more risky your development plans are.
Should you try to minimize variation?
Yes, for early studies
Easier to detect signals
Proof of concept trials
No, for later studies
Easier to generalize results
Pragmatic trials
Speaker notes
It is a bit of a generalization, but most researchers try to avoid variation in early studies. By early studies, I mean studies of therapies that have not yet been extensively tested in a broad range of settings. Less variation means that there is a greater chance to detect signals. You remove variation by using very strict entry criteria on who can get into the study. You remove variation by tightly controlling what the patient is allowed to do (e.g., no concommitant medications). You remove variation by tightly standardizing the delivery of the intervention and the assessment of the outcome. You reduce variation by removing patients who deviate from the research protocol requirements.
These are known as proof of concept trials. If a new therapy cannot succeed even under the tight controls, there is no point in studying it futher. But success in a tightly controlled environment does not guarantee success in the real world.
If you are planning a trial that comes after many similar trials, you actually may want to encourage variation. Broaden the inclusion criteria so that the patients in the trial look no different than the patients you see every day in your clinic.
Standard deviation
\[S = \sqrt{\frac{1}{n-1}\Sigma(X_i-\bar{X})^2}\]
At least one alternative formulas.
Speaker notes
The standard deviation is a commonly used measure of how spread out the data is. The formula is a bit messy, but if you look carefully at it, you will see that it is a measure of how far each individual value is from the overall mean.
Now, maybe you’ve seen or used a different formula. Don’t worry about it. In a short course like this, I won’t ask you to calculate anything as tedious as a standard deviation. Let the computer do all of the work.
The bell shaped curve
Does your variation follow a bell shaped curve?
Values in the middle are most common
Frequencies taper off away from the center
Symmetry on either side
A bell shaped curve = better characterization of variation
Speaker notes
Much variation in the real world follows a bell shaped curve, alternately called a normal distribution. You can assess whether you have a bell shaped curve using a histogram. Look for values in the middle being most common. The frequencies should taper off slowly as you moved away from the middle. The histogram should have symmetry. The left side of the histogram should be roughly equivalent to the right side of the histogram.
Not a bell shaped curve
Figure 12: Bimodal histogram
Speaker notes
Here’s a histogram that shows a bimodal distribution. The frequencies are not highest in the center of the data. This is not a bell shaped curve.
Not a bell shaped curve
Figure 13: Skewed histogram
Not a bell shaped curve
Figure 14: Uniform histogram
Speaker notes
Here’s a histogram that shows a symmetric distibution, but the frequencies do not taper off as you move away from the center. This is not a bell shaped curve.
Not a bell shaped curve
Figure 15: Heavy-tailed histogram
Speaker notes
Here’s a histogram that shows a symmetric distibution, but the frequencies taper off at first, but then flatten out. This is called a heavy tailed distribution and it tends to produce outliers, extreme values, on both sides. This is not a bell shaped curve.
A bell shaped curve (finally!)
Figure 16: Bell-shaped histogram
Speaker notes
Here’s a histogram that shows a symmetric distribution, with the most frequent values in the center and frequencies that taper off on either side. This is a bell shaped curve.
Plus or minus one standard deviation
Figure 17: Percentage within one s
Speaker notes
This shows the bell shaped curve with the data within one standard deviation of the mean highlighted in gray. Roughly 68% of the data lies within one standard deviation of the mean. This is only true if the variation follows a bell shaped curve.
